''For Lebesgue's lemma for open covers of compact spaces in topology see
Lebesgue's number lemma''
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Lebesgue's lemma is an important statement in
approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...
. It provides a bound for the projection error, controlling the error of approximation by a linear subspace based on a linear projection relative to the optimal error together with the
operator norm of the projection.
Statement
Let be a
normed vector space, a subspace of , and a
linear projector on . Then for each in :
:
The proof is a one-line application of the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
: for any in , by writing as , it follows that
:
where the last inequality uses the fact that together with the definition of the
operator norm .
See also
*
Lebesgue constant (interpolation)
In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree ...
References
*
{{DEFAULTSORT:Lebesgue's Lemma
Lemmas in analysis
Approximation theory